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Finding riccati solution of A*X+A'*X+X*W*X+Q

  1. Jul 6, 2004 #1

    gs

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    in finding riccati solution of

    A*X+A'*X+X*W*X+Q that is

    X which stabilises A+W*X(real parts of eigen values are <0) ,it’s existence can
    Found out by
    Eigen values of Hamiltonian matrix H given by


    H MATRIX=
    !A W!
    !Q -A!
    because we have the relation

    EIGEN VALUE OF H ARE GIVEN BY= EIGENVALUES OF (A+W*x)& - (A+W*x);

    In text it is stated as if there is no eigen values of H are on imaginary axis then X exists

    Means it can have in real parts of ( eigen values can be >0)

    This can be possible
    If A+W*x has negative real parts

    And also A+W*x has positive real parts in which it is un stable

    If it is so how can we say that just H matrix not having eigen values on imaginary axis is
    Sufficient for X toexist
    Can any one explain me about this
    Thanking you
     
  2. jcsd
  3. Jul 8, 2004 #2
    Shouldn't hamiltonian be a hermite operator [tex]H=H^{\dagger}[/tex]. Then you would have W=Q*.
     
    Last edited: Jul 8, 2004
  4. Jul 9, 2004 #3

    gs

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    ya itis right but how it explains the existence of X
     
  5. Jul 9, 2004 #4
    If you could rephrase the text I might help you more.
     
  6. Jul 9, 2004 #5

    gs

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    my point is to if H has real parts of eigen values greater than zero;which may be due to either A+W*X is having eigen values greater than zero;ordue to -(A+W*X)in which
    case A+W*X has negative eigenvalues .hence we cannot say whether X exists or not just by looking at the any eigen values on imaginary axis ;means this condition for existance of X is not sufficient ,which is my understanding but in text it stated is sufficient ,i want to know how can it.
     
  7. Jul 9, 2004 #6
    Well, sorry I cannot help you with that.
     
  8. Jul 12, 2004 #7

    gs

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    thing is the relation of eigen values of H and eigen values A+W*X is valid only for X Stable.hence it is sufficient
     
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